Socioeconomic Planning Sciences 34 (2000) 199-218
PARTITIONING INPUT COST EFFICIENCY INTO ITS
ALLOCATIVE AND TECHNICAL COMPONENTS. AN
EMPIRICAL DEA APPLICATION TO HOSPITALS*.
Jaume Puig-Junoy
Universitat Pompeu Fabra
Department of Economics and Business
Trias Fargas 25-27
08005 Barcelona (Spain)
Fax 34.3.542.17.46
e-mail jaume.puig@econ.upf.es
(*) This paper is part of a research supported by the Comisión Interministerial para la Ciencia y la
Tecnología del Ministerio de Educación y Ciencia (CICYT)under contract SEC94-0192 and by the
BBV Foundation. I am grateful to G. López from CRES and to two anonymous referees for helpful
comments that have substantially improved the paper.
.
PARTITIONING INPUT COST EFFICIENCY INTO ITS
ALLOCATIVE AND TECHNICAL COMPONENTS. AN
EMPIRICAL DEA APPLICATION TO HOSPITALS.
ABSTRACT
The study presents an empirical analysis of best practice production and cost
frontiers for a sample of 94 acute care hospitals by applying Data Envelopment
Analysis (DEA) and a regression model, in a two-stage approach. This paper
contributes to the DEA and efficiency measurement literatures by adding results from
a homogeneous method of partitioning cost efficiency into its allocative (or price)
and technical components, and by decomposing technical efficiency into scale,
congestion and pure technical efficiency. Allocative efficiency is calculated using a
DEA assurance approach. It introduces constraints with lower and upper bounds on
the admissible values of weights of the CCD DEA model that computes technical
efficiency. We thus obtain scores unbiased by the lack of precise information on
input prices. In the second stage, a log-regression model is employed to test a number
of hypotheses involving the role of ownership, market structure, and regulation in
terms of differences amongst the various efficiency concepts measured. Results
highlight the relevance of market concentration and public finance in explaining
these differences.
KEYWORDS: Hospital Performance, Cost Efficiency, Allocative Efficiency,
Technical Efficiency, Data Envelopment Analysis, Assurance Region.
1
1. INTRODUCTION
The purpose of this paper is to obtain empirical and complementary measures of hospital
performance rooted in the principles of production economics, and to evaluate the factors that
contribute to performance. The method is applied to 94 acute care hospitals operating in the
1
context of a National Health Service in Catalonia (Spain) . Assessing performance is a necessary
step in the design and implementation of privatization of ownership and management policies,
and in fostering competition and other deregulating measures in health and hospital services. In
this regard, health care purchasers in all systems are now seeking ways to improve hospital
efficiency.
Hospital performance is proxied in this paper using measures of Farrell's [1] definition of
technical and allocative efficiency. These are partial, but theoretically rooted, indicators of
hospital performance. A hospital is said to be technically efficient if a reduction in any input
requires an increase in at least one other input or a decrease in at least one output. A hospital is
allocatively inefficient if it does not select the optimal mix of inputs given the available
technology and the input prices it faces. Technical efficiency has been advocated as a measure to
compare performance of firms having different ownership regimes or legal statuses. It is
particularly useful in evaluating the performance of public sector and nonprofit activities, which
are predominant in the hospital sector. Technical efficiency may be achieved independently of
allocative efficiency.
1.1 Measuring hospital efficiency.
Empirical measurement of inefficiency has been accomplished using two classes of
methodologies: stochastic parametric regression-based methods and nonstochastic nonparametric
mathematical programming methods. Data envelopment analysis (DEA) is the most used family
of linear programming models.
A number of papers have measured hospital efficiency on the basis of the best-practice
frontier by using both methodologies. Inefficiency provided by hospital cost frontiers is the result
of technical and allocative inefficiency combinations in unknown proportions [2]. Eakin [3] is an
1
Catalonia is a region with six million inhabitants. The hospital system in Catalonia may be summarized as
acting in a National Health Service with the following measures: 99% of population with public insurance,
73% public financing and 39% public production.
2
exception, computing allocative efficiency scores. Some advances in frontier regression analysis
allow one to obtain differentiated measures of technical and allocative inefficiencies by
introducing restrictions equalizing marginal productivity ratios and price ratios in the cost
function. Nevertheless, computational difficulties in panel data aside, some problems exist in
ruling out X-inefficiency when separating both types of inefficiency in cost frontier regression
analysis. These are due to the assumption that maximizing behaviour is present [4] since it uses
the so-called Shepard cost share equations to estimate model parameters. In response to this
situation, several DEA models are proposed here to partition cost efficiency into its allocative and
technical components within a multiple input multiple output production process.
An increasing number of researchers have applied DEA to hospital efficiency analysis.
Some recent examples include: Burgess and Wilson [5], Valdmanis [6], Ozcan and Luke [7],
Magnussen [8], and Dalmau and Puig [9]. The hospital DEA literature has restricted its attention
to technical efficiency, although cost-minimizing efficiency includes both technical and
allocative efficiency. To our knowledge, only two papers calculate hospital allocative efficiency
[10,11] using nonparemetric models. Calculation of allocative efficiency requires accurate
information on prices of inputs. Morey, Fine and Loree [10] and Byrnes and Valdmanis [11] use
average prices to calculate allocative efficiency for public and nonprofit hospitals in Californa in
the period 1982-1983. However, in both cases, average input prices involve an unreasonably
wide range of variation between hospitals, which is not justified by the authors. In this regard, we
suggest that less quality of cost data are available than physical input data in self-reported sources
of information. As might be expected, a major difficulty is encountered in securing the price
information needed to implement the concept of allocative efficiency.
This paper's contribution to the DEA applications literature involves the use of this method
to derive both allocative and technical efficiency scores for hospitals, thus overcoming the
traditional confinement to technical efficiency in earlier efforts. The use of DEA provides the
opportunity to partition cost efficiency into its allocative and technical efficiency (and the latter
into pure technical, congestion and scale efficiency), and to subsequently obtain comparable
measures of the different theoretical efficiency concepts. Results should cast light on the relative
importance of the different types of inefficiency for hospitals under analysis. Additionally, a
DEA assurance approach is applied to the calculation of allocative efficiency in order to obtain
scores unbiased by the lack of precise information on input prices.
3
1.2 Explaining variations in hospital measured efficiency.
According to Pestieau and Tulkens' [12] theoretical and empirical revision, three categories
of factors might be distinguished in assessing and explaining the performance of public and
nonprofit enterprises: ownership (and firm objectives), competition, and regulation. In order to
assess the expected effects of projected and in-course hospital policies, it is thus of crucial
importance to ascertain the potential impact of ownership, market structure and regulation on the
explanation of differences in efficiency scores.
Evidence from empirical analyses of hospital inefficiency using DEA several times on the
same set of data, Grosskopf and Valmanis [13] and Valdmanis [6] suggest that public hospitals
are more technically efficient than are nonprofit and private ones. Register and Bruning [14], also
using DEA, found no differences between nonprofit and public hospitals when comparing
technical efficiency. Ozcan, Luke and Haksever [15] and Ozcan and Luke [7] observed that US
government hospitals tend to be more efficient, and for-profit hospitals less efficient, than other
hospitals. Chirikos and Sear [16] conclude that for-profit hospitals are technically less efficient
when they perform in less competitive markets.
A number of earlier papers have documented a positive relation between costs per
admission or per patient-day and more competitive markets [e.g. 17,18], usually attributed to the
effects of nonprice competition. Nevertheless, only a few of these papers have addressed the
relation between efficiency and competition. Recent empirical research estimating a frontier cost
function found weak evidence to sustain the notion that competition from other hospitals is
related to inefficiency [2, 3]. A positive relation between competition and higher average cost or
cost inefficiency does not necessarily imply technical inefficiency. It might, for example, be a
case of exclusively allocative inefficiency, or both technical and allocative inefficiency in
different, unknown proportions. Two studies explicitly address the effect of competition on
technical inefficiency by explaining differences in DEA scores. Register and Bruning [14] did not
find any relation between DEA scores and market concentration. Chirikos and Sear's [16] results
showed that inefficiency scores are higher in markets with more vigorous inter-hospital
competition, the relation being more intense in highly competitive markets.
Analysis of the relation between regulation and hospital efficiency has focused nearly
exclusively on Medicare Prospective Payment System evidence (PPS). Zuckerman et al [2] thus
found that profit rates are significantly higher among relatively less cost inefficient hospitals
4
subject to PPS. Chirikos and Sear [16] found no significant relation between technical efficiency
and an index of early PPS pressures.
In this paper, an evaluation of the effects of observed present market structure, ownership
and regulation on hospital allocative and pure technical efficiency for 94 Catalan acute care
hospitals is developed. In reference to the relation between hospital performance and factors
explaining performance, this paper adds to the preceding literature in three aspects. Firstly, it does
not restrict attention to larger or urban hospitals since all acute care hospitals are considered. It
uses a Herfindahl-Hirschman index [19] of concentration calculated for every hospital using
patient origin data, and it expands evidence to hospitals in a European National Health Service
context. Secondly, it encompasses the analysis of a wider range of environmental variables
considered simultaneously as factors explaining efficiency, and it also considers some control
variables for efficiency scores. Ratios partially measuring inefficiency are ruled out as factors
explaining efficiency (i.e. occupancy rate, length of stay, etc.). And, thirdly, it sheds separate light
on the effects of environmental variables on allocative and technical efficiency (rather than on
average production/cost functions).
The paper is organized as follows. Section 2 lays out the general framework for the
application of Data Envelopment Analysis to the measurement of cost and technical efficiency.
Variable definitions and descriptions are presented in Section 3. Section 4 presents DEA
allocative and technical inefficiency results. A regression analysis of the DEA efficiency scores is
presented in Section 5, while section 6 concludes.
2. THE PERFORMANCE EVALUATION METHODOLOGY
As noted previously, hospital performance is proxied in this paper by allocative and
technical efficiency. In this section, we provide definitions of efficiency used and their methods
of measurement.
Efficiency definition.- To characterize production technology relative to which efficiency is
N
measured, each hospital uses variable inputs x = (x1, ..., xN) ∈R + to produce variable outputs y =
(y1, ..., yM) ∈RM+. Inputs are transformed into outputs using a technology that can be described by
the graph GR = {(x,y) : x can produce y}. Corresponding to the graph, there is a family of input
5
sets L(y) = {x : (x,y) ∈ GR}, y ∈ RM+. Input sets are assumed to be closed and bounded above,
and to satisfy strong disposability of inputs. The input sets contain isoquants Isoq L(y) = {x : x ∈
L(y), θx ∉L(y), θ ∈ (0,1)}, y ∈ R +. Also corresponding to the graph of the technology is a
M
family of output sets P(x) = {y : (y,x) ∈ GR}, x ∈ RN+. Output sets are assumed to be closed and
bounded above, and to satisfy the properties of convexity and strong disposability of outputs.
A Farrell-Debreu radial measure of the technical efficiency of input vector x in the
production of output vector y is given by: TE(x,y) = min {θ : θx ∈L(y)}, where θ = 1 indicates
radial technical efficiency and θ < 1 shows the degree of radial technical inefficiency. According
to Farrell's concept, the cost efficiency of a hospital using input vector x to produce output vector
y when input prices are w is measured by the ratio of minimum cost to actual cost: CE (x,y,w) = c
T
(y,w) / w x, where c(y,w) is the cost function (the minimum expenditure required to produce y
when input prices are w), and where CE (x,y,w) = 1 indicates cost efficiency and CE (x,y,w) < 1
shows the degree of cost inefficiency.
Efficiency measurement.- Assuming strong input and output disposability, the input cost
efficiency measure (CE) may be decomposed into its input allocative efficiency (AE), scale
efficiency (SE), input congestion (C), and pure technical efficiency (PTE) components [20, p.
80]: CE (x,y,w) = AE(x,y,w) ⋅ SE(x,y) ⋅ C(x,y) ⋅ PTE(x,y). As Färe, Grosskopf and Lovell [20]
state "the input cost inefficiency must be due to selection of the wrong input mix, to the adoption
of an inefficiently small or large scale, to input congestion, or to purely technical inefficiency".
Scale inefficiency thus occurs because the hospital is not operating at the scale of operation
consistent with long-run competitive equilibrium. Also, technical efficiency (TE) is defined as the
product of the scale efficiency, input congestion, and pure technical efficiency components:
TE(x,y) = SE(x,y) ⋅ C(x,y) ⋅ PTE(x,y). The Farrell input allocative efficiency of a hospital is
measured as the ratio of cost efficiency to overall technical efficiency: AE(x,y,w) = CE(x,y,w) /
TE(x,y), where AE (x,y,w) = 1 indicates input allocative efficiency and AE (x,y,w) < 1 shows the
degree of input allocative inefficiency.
0
0
0
Let's assume the hospital under evaluation as having data (x , y , w ), and consider the
i
N
input-oriented CCR DEA model [21] in the primal (envelopment) formulation, where x ∈R + and
yi ∈RM+, and i = 1 ... I, where I indicates the number of hospitals in the sample:
TE (x0,y0) = min θ
6
θ,λ
θx - Xλ ≥ 0
0
subject to
-y + Yλ ≥ 0
0
λ≥0
i
i
where X is an NxI input matrix with columns X , Y is an MxI output matrix with columns y , and
λ is an Ix1 intensity vector. The optimal value of θ provides a technical efficiency measure of the
hospital under evaluation. Input-oriented radial efficiency requires uTyo = θ = 1. A hospital is
judged to be technically inefficient if, at optimum, θ < 1, and technically efficient if, at optimum,
θ = 1. The input-oriented CCR DEA model incorporates the assumption of constant returns to
scale in production.
Banker, Charnes and Cooper [22] (BCC) generalized the CCR formulation to allow
variable returns to scale. The input-oriented BCC DEA model computes, exclusively, a pure
technical efficiency measure (W) by introducing an additional restriction to the input-oriented
CCR DEA model: eTλ = 1, where eT is an Ix1 row vector of ones. This pure technical efficiency
measure is obtained under the restriction of weak input disposability but allows for variable
returns to scale. The above decomposition of input cost efficiency requires PTE to be computed
by relaxing the strong input disposability restriction, to allow for an input congestion component.
The congestion component is due to production on a backward-bending segment of the isoquant
that is in the region where marginal product is negative. Pure technical efficiency with weak
disposability of inputs may be computed from the following problem in the primal (envelopment)
formulation:
PTE (x ,y ) = min θ
θ,λ,σ
0
subject to
0
θσx0 - X λ = 0
0
-y + Y λ ≥ 0
eTλ = 1
0<σ<1
λ≥0
Then, the congestion measure is obtained as: C = W / PTE.
Input cost efficiency can be measured for hospital (y0,x0,w0) by solving the following linear
programming problem, under the assumption of constant returns to scale [20]:
7
0
0
0T
c (y ,w ) = min w x
x
subject to
x - Xλ ≥ 0
-y + Yλ ≥ 0
o
λ≥0
The proposed cost efficiency DEA model has similar constraints to those of the CCR DEA
model, but it differs in the objective function (cost) and in the number of input units (x), which
must be determined by the model. The cost efficiency linear problem previously defined is
similar to the applied models of Morey, Fine and Loree [10] and Ferrier and Lovell [23], which
2
differ in constraints relating to returns to scale assumptions , and of Byrnes and Valdmanis [11].
However, usually data requirements to compute allocative efficiency cannot be satisfied because
prices of hospital inputs are not accurately observed, thus severily limiting actual applications.
When it is not possible or difficult to ascertain exact knowledge of prices, input cost efficiency
cannot be measured. One solution to this problem involves a recourse to average prices, this
being this the approach used in hospital applications [10,11]. Two problems may appear in this
case. First, average prices can be a misrepresentation if: (i) inputs are not completely
individualized and defined with some degree of aggregation in the DEA problem; (ii) there exists
some variability during the period under analysis; and (iii) there are reasonable doubts about data
reliability on resource quantity or costs, especially when data come from self-reported sources of
information. Second, prices can be (and often are) subject to variation in very short periods so
that additional choices and assumptions are involved concerning their pertinence.
An alternative to computing input cost efficiency, as suggested by Cooper, Thompson and
Thrall [24], is to introduce constraints with lower and upper bounds on the admissible values of
weights of the CCR DEA problem. It is known that as increasingly severe constraints are placed
on weights, so the measure of efficiency derived moves from one of relative technical efficiency
to one of relative overall efficiency. In this way, knowledge of exact price information could be
replaced by knowledge of upper (UL) and lower (LB) bounds within which relative prices are
expected to vary using the Assurance Region (AR) approach first developed by Thompson et al
[25]. The AR approach introduces separate linear homogenous restrictions on input (and output)
2
Morey, Fine and Loree [10] define a constraint not allowing decreasing returns to scale (eTλ ≥ 1)in
hospital cost efficiency estimation. Ferrier and Lovell [23] limit efficiency measurement in banking to
variable returns to scale (eTλ = 1).
8
weights of the multiplier (dual) DEA problem. We specify a Cone-Ratio Assurance Region (CRAR)[26]. Let the h-tieth input (Xh) be a numeraire for the inputs. Then, an assurance region may
be specified as follows: ai⋅υh ≤ υi ≤ bi⋅υh, i=1,...,N, where ai=LBi/UBh and bi=UBi/LBh , and where
υi is the weight given to the i-th input.
Usually, the application of CCR and BCC DEA models place no constraints on weights
attributed to each input (υ) and each output (u) in the dual (multiplier) problem, thus allowing
absolute weight flexibility. Then, outlier or extreme units will be automatically classified as
technically efficient units by assigning a zero weight (i.e., weights of very small magnitude) to
some of the inputs or outputs. This represents a contradiction in itself because if such inputs or
outputs are not important, they would not be included in the analysis [27, page 218-20]. Absolute
weight flexibility may result in an overestimation of technical efficiency, and, consequently, may
also result in an overestimation of allocative inefficiency when cost efficiency is decomposed into
its technical and allocative components. The solution could be the imposition of restrictions on
the weights in the multiplier DEA problem. This would suggest the formulation of value
judgements about the relative importances of the inputs and/or outputs. In this paper, however,
technical efficiency is computed from CCR and BCC unbounded DEA models, and a DEA
Assurance Region approach which constraint input weights to compute allocative efficiency
scores. We do this for two reasons. First, using simulated data from a Cobb-Douglas production
function with constant returns to scale, Pedraja et al [27] showed that, using simulated data from
Cobb-Douglas production function with constant returns to scale, as sample increases the
3
overestimation of efficiency is lowered in unbounded DEA models . Importantly, overestimation
represents a problem only when the size of the sample is relatively small. And, second, the only
implicit hypothesis in the unbounded CCR and BCC DEA models is that the weights adopted by
the hospital under scrutiny are acceptable. This may be an acceptable principle when no
judgement on allocative efficiency is required. When constraints on the weights are introduced,
the measure of efficiency derived moves from one of relative technical efficiency to one of
relative overall efficiency, confusing the partition of input cost efficiency into its allocative and
technical components.
3. DATA AND VARIABLES
3
Note that DEA models always overestimate efficiency when the production function is convex.
9
We now apply the DEA models defined previously to calculate technical efficiency, pure
technical efficiency, scale efficiency, congestion efficiency and cost efficiency for 94 Catalan
4
acute care hospitals . Resolution of the implied mathematical programming problems requires
definition of input and output variables, and input prices. Input and output variables were selected
among those that had been used primarily in the DEA hospital efficiency literature. One of the
most important issues in DEA applications is that computed efficiency scores may be influenced
by the model specification of input and output variables. The magnitude of this problem has been
illustrated for output specification in hospital applications by Magnussen [8]. Input and output
sets used in this paper are very close to those more often employed in the literature on hospital
efficiency. (In a previous paper [9], some alternative model specifications showed high
correlations amongst the obtained efficiency scores.)
Output is defined in this paper as health services or intermediate outputs (throughputs).
Eight separate direct hospital outputs were specified: Case-mix adjusted discharged patients (Y1);
In-patient days in acute and subacute care services, except intensive care units (medicine,
surgical, obstetrical, gynaecological and paediatric services) (Y2); In-patient days in intensive
care units, including intensive neonatal and burn units (Y3); In-patient days in long-term
5
(psychiatric , long stay, and tuberculous services) and other hospital services (Y4); Surgical
interventions (Y5); Hospital daycare services (Y6); Ambulatory visits (Y7); and Resident
physicians (Y8). Selected output variables represent both in-patient (admissions and in-patient
days) and out-patient hospital services (visits and day care), as well as teaching activities
(residents).
On the input side, four variables representing resource consumption are defined: Full-time
equivalent (FTE) physicians, including residents (X1); FTE nurses and equivalents (X2); FTE
4
Data (circa 1990) came from the Estadística de Establecimientos Sanitarios con Régimen de Internado, a
survey conducted annually by the Department of Health and Social Security of the autonomous
government in cooperation with the Instituto Nacional de Estadística (National Institute of Statistics). This
set of data is the only one that allows one to employ accurate information on case-mix of public, nonprofit
and private hospitals as well as a Herfindahl-Hirschman index of market concentration for the Catalan
hospital market previously estimated in Dalmau and Puig [9].
5
All hospitals included in the study are acute care hospitals in order to make comparisons between
homogeneous decision making units. Although being acute care hospitals, in some cases, they have small
psychiatric facilities.
10
other non-sanitary personnel (X3); and In-patient beds (X4). A summary of all variable definitions
appears in Table 1.
[ Table 1 ]
Note that the first three inputs are labour while the last one is a proxy for net capital assets,
as suggested by Grosskopf and Valdmanis [13]. Input and output quantities and input prices
represent average measures during the period under study. Descriptive statistics for all input and
output variables appear in Table 2.
[ Table 2]
The Assurance Region (AR) approach to cost efficiency requires the introduction of
restrictions on the input weights. Because input prices are imperfectly known, in the AR
approach they are replaced by the bounds within which prices are expected to vary. Specification
of upper and lower limits ("assurance region") of input prices used in this paper come from
market prices and expert opinion. First, information on wages for full time hospital physicians,
nurses and other personnel were obtained from those officially established for hospitals belonging
to Social Security, and from public agreements between trade unions and the principal hospital
6
associations . Second, this information has been revised by three experts in order to estimate
potential variations around obtained wages. Third, upper and lower limits for the cost per inpatient bed were established from average prices revised by expert opinion. And fourth, lower
and upper bounds were set for all inputs allowing for possible variations in relative input prices
from 1990 to 1997. This, we felt, helped to guarantee the continued relevance of the prices used.
7
Indeed, major changes have not been detected during this period .
4. EVALUATING HOSPITAL PERFORMANCE
Under a DEA formulation, the performance of a hospital is evaluated in terms of its ability
to contract its input vector given its output vector subject to constraints imposed by best observed
6
Consorci Hospitalari de Catalunya and Unió Catalana d'Hospitals.
11
practice. For the hospital being evaluated, the positive elements of λ identify that set of
dominating hospitals located on the constructed frontier (best observed practice), againts which
the hospital is evaluated. Given the input and output data sets, DEA identifies the efficient
hospitals or constructs a linear combination of them in the sample as those representing the best
observed practice. The constructed production frontier is then used as a reference for inefficient
hospitals. DEA is a nonparametric approach, which is less prone than econometric approaches to
misspecification of the functional form of the production function (the relation between inputs
and outputs). The way in which DEA falls short of modelling a hospital depends primarily on the
appropriateness and measurement of the inputs and outputs. However, our cost and technical
efficiency measures are relative measures in the sense that inefficient hospitals are compared with
the best observed practice in the sample of hospitals being analyzed. A change in the analyzed
hospitals may thus shift the frontier and result in lower efficiency scores for some hospitals. What
can be predicted is that hospitals identified as inefficient may decrease their inputs without
decreasing their outputs, until they reach the frontier, given the sample of references and the input
and output vectors.
DEA models presented in previous sections allow us to compute relative measures of cost,
allocative, technical, pure technical, scale and congestion efficiencies. Table 3 summarizes the
average scores for the five efficiency concepts measured in this paper.
[ Table 3 ]
Results show an average technical inefficiency of 10.1%. That is to say, hospitals would
need to lower inputs 10.1% if all were operating on the production efficiency frontier. The
average overall efficiency scores range from 0.545 to 1. Pure technical inefficiency scores show a
lower level of inefficiency, the average being 2.9%. Average scale inefficiency is 4.6% while
congestion efficiency is 2.8%. For technical efficiency, the percentage of hospitals operating on
the frontier is 36.2. The average efficiency score for nonfrontier hospitals is 0.841, implying that
non-efficient hospitals use, on average, 18.9% more inputs per unit of output than do efficient
hospitals. According to the pure technical efficiency criterion, 69 of the hospitals operate
efficiently, with an average efficiency score of 0.971 for nonfrontier hospitals. The distribution of
all scores is summarized in Table 4.
7
The values used in this paper are as follows, for h=1: a1 = 0.79, a2 = 0.48, a3 = 0.34 and a4 = 0.16;
12
[Table 4 ]
The decomposition of technical efficiency shows that, on average, scale inefficiency
accounts for over 45% of technical inefficiency. However, only in seven of the 25 hospitals
showing pure technical inefficiency is scale inefficiency the primary source of technical
inefficiency. Nearly two-thirds of the inefficient hospitals, according to TE, are operating under
decreasing returns to scale. However, 23 other inefficient hospitals show increasing returns to
scale.
Average cost inefficiency for all hospitals under analysis is significantly higher than overall
technical inefficiency. Hospital cost is, on average, 24.5% higher than would be necessary if all
hospitals were operating on the best-practice cost efficiency frontier. Cost efficiency scores range
from 0.395 to 1. Only 15 hospitals (16%) are on the cost frontier, i.e, operating at minimum cost.
The average cost efficiency score for nonfrontier hospitals is 0.766, implying that excess cost in
these hospitals is 23.4%.
Cost efficiency is decomposed into allocative (or price) efficiency and technical efficiency.
A technically efficient hospital may show cost inefficiency because, given input prices, it does
not minimize cost. According to our results, average allocative inefficiency is 10.9%, ranging
from 47.2% to zero. While a hospital may be technically efficient, it may be at the same time
allocatively inefficient, and vice versa. Then, the primary source of inefficiency is allocative
inefficiency. Given the lower and upper bounds of input prices, cost inefficient hospitals employ
a non-optimal mix of inputs. Here, their costs are 30.5% higher than the cost-minimizing level.
Table 5 shows summary statistics on the possible savings as obtained from the analysis of
overall technical efficiency. For each input, more than half the hospitals show no possible
savings; the other half have input slack. Possible savings represent a complementary dimension
of technical inefficiency. Table 5 suggests that if it were possible for the 60 inefficient hospitals
to perform like the best-practising 34, savings of 7.38 % in the number of physicians would be
possible. The larger slacks occur in the nurses input, which could be reduced by more than 14%
in inefficient hospitals. At the same time, potentially increased outputs can be observed. Acute
and b1 = 1.30 , b2 = 0.75 , b3 = 0.55 and b4 = 1.45 .
13
care patient-days and ambulatory visits thus show a very low potential for increase as do the
number of residents and discharged patients. Potential increases arise in the augmented number of
surgical interventions, days of stay in intensive care units, and the number of daycare services,
the latter could be more than doubled in the inefficient hospitals.
[Table 5]
8
To discriminate between relatively efficient hospitals, a Cross Efficiency Matrix has been
employed. In this structure, the number an efficient hospital appears in the peer groups of
inefficient hospitals. An indication of robustly efficient hospitals is the number that appear in the
peer groups of inefficient ones. Five of the 34 hospitals showing technical efficiency appear in 30
or more comparison groups of inefficient hospitals (more than half of them). On the other side,
seven efficient hospitals do not appear in any comparison group of inefficient hospitals.
5. AN ECONOMETRIC ANALYSIS OF PERFORMANCE
What causes a hospital to produce by using more than the minimum quantity of inputs for a
specific vector of outputs? What causes a hospital to spend more than the minimum for a specific
vector of outputs? What are the factors associated with efficiency?
In order to determine the influence of environmental variables on efficiency, we adopt a
two-stage approach. Let zj ∈ R+, j = 1, ..., I be a discrete or continuous environmental variable. zj
represents variables over which the hospital has no control during the time period under
consideration; zj and the input variables should be uncorrelated. In the first stage inefficiencies are
calculated using a DEA model in which the environmental variables are ignored. In the second
stage variation in calculated efficiencies is attributed to variation in operating environments by
means of a regression model. It is hypothesized that the DEA efficiency measures are some
function of the vector z and a random disturbance εi:
8
A Cross Efficiency Matrix is a table that conveys information on how a hospital's relative efficiency is
rated by other hospitals using their DEA optimal weights.
14
θ i = f( zij ) ε i
Some authors, such as Rosko et al [28], Chilingerian [29] and Kooreman [30], have
conceptualized DEA efficiency scores as a censored normal distribution; that is, those values of
the dependent variable in the regression model above a threshold are measured by a concentration
of observations at a single value. We would thus suggest that ordinary least squares, as used by
Chirikos and Sear [16], was not an entirely appropriate method. Some authors [9, 30] have
therefore concluded that a censored Tobit model is appropriate in order to avoid biased estimates
from ordinary least squares. The Tobit model is based on normally distributed latent variables.
However, DEA scores do not fit the theory of censored sampling that gives rise to Tobit models;
i.e., accumulation of sample observations at the highest level of efficiency. Importantly, Tobit,
and also Probit, estimates are inconsistent in the cases of non-normality of error terms and/or
error term heterogeneity. To overcome this problem, several alternatives have been proposed in
the literature: Luoma et al [31], for example, used a test statistic sensitive for linear fit; in
particular, the type of heteroscedasticity which is related to fit and excess skewness of the error
terms. González and Barber
[32] estimated their model assuming different probability
distributions of disturbance. Burgess and Wilson [33] removed the censoring problem directly
through the addition of information on the distance of each observation from all other
observations in the sample.
Further to these studies, Banker and Johnson [34] proposed a consistent, non-maximumlikelihood estimator in an empirical application based on the theoretical conditions established by
Banker [35]. Given the implausibility of normally distributed latent efficiency as required by
Tobit models, and according to Banker and Johnson [34] we make the assumption that
inefficiencies are log-normally distributed, and define the following transformations:
θ i = 1 / θ i -1+ ω
where ω is some very small number. Then, inefficiency can be posited to be a multiplicative
function of the explanatory variables and random error term:
θ i = β 0 Π j z βj eν
j
15
i
where βj and γj are parameters capturing the relationships between the explanatory variables and
input and output inefficiency, respectively; and exp{υi} is a random error term that is assumed to
be independently and identically distributed and log-normally distributed with mean 1.
As noted earlier, factors explaining the performance of hospitals, as measured by
productive efficiency, may be conceptualized in three categories: ownership, market structure and
regulation. In order to test the empirical impact of market concentration on efficiency it is
necessary to consider other relevant factors. Partial tests, as those of Valdmanis [6] and limited to
the effect of ownership, may present statistical biases due to possible misspecifications of the
regression model.
Ownership is considered here by classifying hospitals in three types: public, nonprofit, and
for-profit. Market structure or competition is proxied by two variables: the Herfindahl-Hirschman
index of local market concentration, calculated for admission data, and the number of competitors
in the local market. The Herfindahl-Hirschman index is specifically used for testing the
hypothesis that there is more efficiency in less concentrated markets. The presence of regulation
influences hospital behaviour through the payment system and patient flows. The proportion of
hospital revenues received from the NHS may be a proxy for the relative importance of
regulation in hospital activities. These monetary flows are influenced by the NHS payment
system as it pertains to patients subject to NHS regulated flows.
The explanatory factors of efficiency are defined here as: Nonprofit hospitals (Z1), Forprofit hospitals (Z2), Herfindahl-Hirschman index of market concentration (Z3), Number of
competitors in the local market (Z4), and Proportion of service revenues from NHS (Z5).
The control variables are present to test the influence of input or output characteristics
omitted or imperfectly measured in the DEA model. Here, they are designed to reflect differences
in severity of treated cases, teaching status and differences in outcome quality. There is no direct
measure available on severity of illness. In this situation, severity is proxied by the number of
surgical interventions with more than one hour per admitted patient. Outcome quality, the direct
measure of which is not available, is proxied here by the proportion of discharged patients with a
recovered health status.
16
Additionally, efficiency scores are controlled for by the potential influence of hospital
dimension (scale economies). We accomplished this by including the number of beds and square
beds. Since overall technical efficiency scores assume that the efficient frontier exhibits constant
returns to scale, hospital size is likely an explaining factor.
According to the preceding arguments, control variables are empirically defined as: More
than one hour surgical interventions per one hundred patients (W1), Teaching status (W2),
Proportion of recovered discharged patients (W3), Number of beds (W4) and Number of square
beds (W5).
Inefficiency scores obtained in this paper were regressed against the explanatory and
control variables. The data were examined for evidence of collinearity and the residuals for
evidence of nonlinearity, nonnormality and heterocedasticity. Correlations between the
explanatory variables were not statistically significant and collinearity was not considered a
problem, except in the case of the Herfindahl-Hirschmann index of market concentration (Z3) and
number of competitors in the local market (Z4). Given the high correlation between these two
variables, and the fact that both measure the degree of market concentration, regression models
were estimated using these two variables separately. Also, nonlinearity was not apparent.
Kolmogorov-Smirnov and Shapiro-Wilks test [36] statistics for normality of the residuals did not
reject normality for the six equations. Finally, the Breusch-Pagan test [36]did not reject
homoscedasticity.
Regression results for DEA efficiency scores as dependent variables are presented in Table
6. The results were obtained using a stepwise method that identifies only one signifcant
explanatory variable for each equation. We found that the explanatory factors do not show
statistical significance in explaining any of the calculated scores. Econometric results indicate
that only the Herfindahl-Hirschman index in the local market (market share)
significant contribution in explaining differences in technical,
presents a
pure technical, scale and
congestion inefficiency scores. In all four cases, as the the level of market concentration
decreases (lower Herfindahl-Hirschman index), the inefficiency level is lower.
Hospitals
operating as local monopolies, i.e., those with very few local competitors, are less technically
efficient than hospitals operating in a more competitive environment. From the econometric point
of view, models with the Herfindahl-Hirschmann index of market concentration (Z3) as an
explanatory variable are preferred to those with the number of competitors in the local market
17
(Z4). These results are in contrast with those obtained from a Tobit estimation in a previous paper
[9]. The earlier work indicated that efficiency was influenced by the number of competitors rather
than by the degree of concentration. Other factors apart from market structure did not show any
relevance in explaining observed levels of technical and pure technical inefficiency.
[ Table 6 ]
Market structure did not appear to influence cost and allocative efficiency in contrast to
technical efficiency. Allocative efficiency was significantly explained by the proportion of
service revenues from NHS (level of public financing). Hospitals with a higher proportion of
revenues from public financing realized a higher allocative inefficiency score than did those with
a lower proportion. That is, a low relative importance on public financing source apparently
contributes to the use of input factors according to their relative prices in a more accurate manner
than in public hospitals or in hospitals receiving a higher proportion of revenues from the NHS.
These results highlight the potential for improving hospital performance through reforms in the
public payment system that provide incentives to minimize cost. It is noteworthy that the greatest
part of inefficiency variation was not explained by the models presented in this section.
In closing, we analyze correlation coefficients of traditional partial indicators of efficiency
with the different efficiency measures (Table 7). Partial ratios considered here are length of stay,
occupancy rate, labour intensity per case, average cost per case, average cost per patient-day, and
proportion of costs covered by grants. These indicators should not be considered as explaining
factors of efficiency scores because they only partially measure the relation between some inputs
and outputs.
[ Table 7 ]
Length of stay shows no statistical correlation with any efficiency score. But then, length of
stay is a poor indicator of efficiency. In contrast, the occupancy rate is positively and highly
correlated with all efficiency scores, except for those of allocative efficiency. The lower the
occupancy rate, the higher the technical inefficiency level. If there is no reason to consider
capacity excess as valued additional output, the occupancy rate may be related to performance.
18
Labour intensity per case is not correlated with pure technical efficiency but is highly and
negatively correlated with scale and allocative scores. This coefficient might indicate, in
consonance with observed potential input savings from cost efficiency scores, that labour is used
in excess relative to other inputs.
The cost per unit (discharge or patient-day) is not correlated with technical efficiency.
Nevertheless, this indicator is negatively related to cost and allocative scores, where both include
consideration of input prices. And, finally, grants partially financing functioning costs are
negatively correlated with score for pure technical efficiency and allocative efficiency.
6. SUMMARY AND CONCLUSIONS
Leibenstein and Maital [37] argued that DEA merits consideration as a primary method for
measuring and partitioning X-inefficiency. In this paper, cost efficiency of acute care hospitals in
Catalonia (Spain) has been analysed by means of an extended version of Data Envelopment
Analysis. The paper has proposed a method to obtain a global measure of cost efficiency
decomposed into its allocative, pure technical, scale and congestion components. In doing so, we
used a DEA Assurance approach to consider the fact that exact knowledge of input prices is
generally difficult.
The main findings of this study may be summarized as follows:
1. Average cost inefficiency in hospital production was 24,5%. Hospital cost was, on
average 24,5% higher than needed if all hospitals were operating on the cost efficiency frontier.
Under the latter conditions, then, cost might be reduced more than a fourth. This inefficiency
level is the result of a 12.2% level of allocative inefficiency, 3.0% of pure technical inefficiency,
4.8% of scale inefficiency, and 2.9% of congestion inefficiency. Allocative inefficiency thus
appears more relevant than does technical efficiency.
2. The degree of market competition (and the number of competitors in the local market)
contributed positively to increased technical efficiency levels. That is, effective or potential
19
competition apparently matters even in a highly regulated hospital market. This conclusion is
more important when many local markets have very few competitors [9].
3. Allocative efficiency was independent of technical efficiency. Thus, privately financed
hospitals (hospitals with a low proportion of service revenues from NHS) may realize higher
allocative efficiency scores in comparison to public and nonprofit institutions (where the latter
have higher proportions of financial resources obtained from public funding).
Problems of DEA frontier estimation are related to the existence of omitted outputs or
inputs, as well as to the assumption of no measurement error or random fluctuations in output.
Although these problems have been managed in this paper through a two-stage approach, this
research might be extended in several ways. Input and output variables might be improved by
taking into account the quality dimension of health. It is essential that the relationship between
quality and input volume be accounted for in the measurement of inefficiency. The method
employed in this paper might be used when estimating allocative and cost efficiencies rather than
limiting attention to technical efficiency. Although we focus on hospital efficiency, here, we did
not determine if patients received an appropriate amount of care.
Additional work on DEA measurement of hospital efficiency should address two main
problems. First, DEA generally ignores the fact that observations in any data set may be subject
to random fluctuations. Deterministic scores should thus be converted into stochastic ones, which
may be obtained through complete panel data and chance-constrained DEA models. And, second,
DEA scores may be biased by measurement problems in the input-output set, which may be
tempered by accurately measuring changes in the severity of illness from admission to discharge.
This would allow for the use of homogeneous patient groups, outcome predictive models
computed at the patient level, e.g. Mortality Probability Model for Intensive Care Units [38], and
measurement of quantity and quality of life for discharged patients. In this ways, we might
improve the measurement of patient characteristics in the input/output set of variables.
20
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23
TABLE 1.
VARIABLE DEFINITIONS
VARIABLE
OUTPUTS
Y1
Y2
Y3
Y4
Y5
Y6
Y7
Y8
DEFINITION
Case-mix adjusted discharged patients
In-patient days in acute care medicine services, except intensive
care units (medicine, surgical, obstetrical, gynaecological and
paediatric services)
In-patient days in intensive care units, including intensive
neonatal and burn units
In-patient days in long-term (psychiatric, long stay, and
tuberculosis) services, as well as other services
Surgical interventions
Hospital daycare services
Ambulatory visits
Resident physicians
INPUTS
X1
X2
X3
X4
Full time equivalent (FTE) physicians, including residents
FTE nurses and equivalents
FTE other non-sanitary personnel
In-patient beds
EXPLANATORY
VARIABLES
Z1
Z2
Z3
Z4
Z5
Nonprofit hospitals
For-profit hospitals
Herfindahl-Hirschman index of market concentration
Number of competitors in the local market
Proportion of service revenues from NHS
CONTROL
VARIABLES
W1
W2
W3
W4
W5
More than one hour surgical intervention per one hundred
patients
Teaching status
Proportion of recovered discharged patients
Number of beds
Number of square beds
24
TABLE 2.
DESCRIPTIVE STATISTICS OF
CONTROL VARIABLES (N=94)
INPUT,
OUTPUT,
EXPLANATORY
MEAN
STDEV
79.6
230.6
132.2
200.0
132.6
335.7
175.2
203.6
3.5
6.6
0.3
14.0
721.0
1765.5
951.5
949.0
OUTPUT VARIABLES
Y1
Y2
Y3
Y4
Y5
Y6
Y7
Y8
6210.4
49868.7
1597.9
6075.1
3761.4
1190.9
49993.5
10.0
6346.0
57925.1
3864.0
13991.0
3431.7
4315.1
71379.5
31.5
224.4
1133.0
0.0
0.0
61.0
0.0
0.0
0.0
32268.5
285308.0
22512.0
95449.0
15050.0
32021.0
520591.0
163.0
EXPLANATORY
VARIABLES
Z1
Z2
Z3
Z4
Z5
0.263
0.474
3139.3
16.3
61.8
0.443
0.502
2917.8
16.2
40.9
0.0
0.0
459.0
0.0
0.0
1.0
1.0
10000.0
36.0
100.0
CONTROL
VARIABLES
W1
W2
W3
W4
W5
22.8
0.05
92.5
200.2
81075.5
18.4
0.23
10.2
203.6
178047.1
0.0
0.0
10.6
14.0
196.0
86.2
1.0
99.9
949.0
900601.0
INPUT VARIABLES
X1
X2
X3
X4
25
MIN
MAX
AND
TABLE 3.
DEA EFFICIENCY SCORES
EFFICIENCY
DEFINITION
CE
MEAN
STDEV
MIN
MEDIAN
% EFFICIENT
0.894
NUMBER
EFFICIENT
15
0.803
0.152
0.395
AE
0.891
0.095
0.528
0.906
15
16.0
TE
0.899
0.124
0.545
0.941
34
36.2
PTE
0.971
0.069
0.584
0.998
69
73.4
SE
0.954
0.094
0.545
0.993
34
36.2
C
0.972
0.066
0.577
0.999
56
59.6
16.0
Note: CE = Cost efficiency; AE = Allocative efficiency; TE = Technical efficiency; PTE =
Pure technical efficiency; SE = Scale efficiency; C = Congestion efficiency. Sample size:
N=94.
26
TABLE 4.
DISTRIBUTION OF DEA SCORES
SCORE VALUE
Less than 0.500
0.500 - 0.599
0.600 - 0.699
0.700 - 0.799
0.800 - 0.899
0.900 - 0.999
1.000
SCORE VALUE
Less than 0.500
0.500 - 0.599
0.600 - 0.699
0.700 - 0.799
0.800 - 0.899
0.900 - 0.999
1.000
CE
Number
3
9
10
19
25
13
15
AE
%
3.2
9.6
10.6
20.2
26.6
13.8
16.0
Number
2
3
7
29
38
15
PTE
Number
1
3
8
13
69
%
2.2
3.2
7.4
30.9
40.4
16.0
TE
Number
5
3
9
19
24
34
SE
%
1.1
3.2
8.5
13.8
73.4
Number
2
3
1
7
47
34
%
2.1
3.2
1.1
7.4
50.0
36.2
%
5.3
3.2
9.6
20.2
25.5
36.3
C
Number
1
2
5
30
56
%
1.1
2.2
5.2
31.9
59.6
Note: CE = Cost efficiency; AE = Allocative efficiency; TE = Technical efficiency; PTE =
Pure technical efficiency; SE = Scale efficiency; C = Congestion efficiency. Sample size:
N=94.
27
TABLE 5.
POTENTIAL SAVINGS DERIVED FROM TECHNICAL EFFICIENCY SCORES
INPUT
X1 PHYSICIANS
X2 NURSES
X3 OTHER LABOUR
X4 BEDS
Hospitals with zero slack
Total slack as % of total
input for inefficient
hospitals
67
49
62
91
7.38
14.12
8.42
0.00
28
TABLE 6
FACTORS EXPLAINING DEA INEFFICIENCY SCORES
INEFFICIENCY
MEASURE
CONSTANT
CE
Z3
Z5
Adjusted
R Square
F
-1.47973
(-0.7199)
0.15153
(2.380)
0.048
5.66
AE
-1.75006
(-9.158)
0.16104
(2.721)
0.065
7.40
TE
-6.4296
(-3.882)
1.14214
(2.274)
0.043
5.17
PTE
-8.4473
(-5.873)
1.15075
(2.638)
0.061
6.96
SE
-6.5797
(-4.432)
1.01253
(2.249)
0.042
5.06
C
-9.6381
(6.714)
1.67810
(3.854)
0.131
14.85
Note: Total observations: N=94. T-Statistics in parenthesis. CE = Cost efficiency; AE =
Allocative efficiency; TE = Technical efficiency; PTE = Pure technical efficiency; SE = Scale
efficiency; C = Congestion efficiency. Z3 = Herfindhal-Hirschman index of market
concentration. Z5 = Proportion of service revenues from NHS.
29
TABLE 7
CORRELATION COEFFICIENTS OF DEA SCORES WITH PARTIAL
INDICATORS OF PERFORMANCE
INDICATOR
CE
AE
TE
PTE
SE
C
Length of stay
Occupancy rate
Labour intensity
per case
Average discharge
cost
Average cost per
patient day
Percentage of cost
covered by grants
0.1732
0.5488b
0.1570
0.1904
0.1147
0.6394b
0.0745
0.3012b
0.0493
0.4576b
0.0708
0.2688a
0.0242
-0.2597
a
-0.2178
b
-0.2924
-0.3032a -0.5716b
0.0516
0.1585
-0.0096
-0.0474
-0.3388b -0.6235b
0.0263
0.1308
0.0165
-0.1045
-0.4273b -0.2775 a -0.3770b -0.4407b -0.2169
0.0170
-0.5647
b
-0.6130
a
(a) P < 0.01; (b) P < 0.001.
Note.- CE = Cost efficiency; AE = Allocative efficiency; TE = Technical efficiency; PTE =
Pure technical efficiency; SE = Scale efficiency; C = Congestion efficiency.
30